Abstract

In this thesis, our first goal is to formulate a generating function and compute its moments alongside the corresponding Hankel determinant. When the latter is nonzero, we will prove that for Painlevé 5, we can construct a Lax pair whose solution is a combination of the solution of the Riemann Hilbert Problem (RHP) and the generating function. An ingredient of that solution, called the Hamiltonian will be used to construct the Tau function which solves the ODE Painlevé V. As such, it will be easy to show that when the Hankel determinant vanishes, the RHP is not solvable, and its zeroes correspond to the poles of the rational solution of the ODE Painlevé V i.e. the Tau function.
On the other hand, an asymptotic analysis will be conducted to prove that the domain of the poles of the rational solution of the ODE Painlevé V (its domain of non analyticity) defines a well shaped region with boundaries on the complex plane as the size of the square Hankel matrix goes to infinity.